Here's my (somewhat cheeky) summary, with comments and evaluation at the end.
Elisabeth, May 6, 1643:
I'm so ignorant and you're so learned! Here's what I don't understand about your view: How can an immaterial soul, simply by thinking, possibly cause a bodily action?
Specifically,
it seems that how a thing moves depends solely on (i) how much it is pushed, (ii) the manner in which it is pushed, or (iii) the surface-texture and shape of the thing that pushes it. The first two of those require contact between the two things, and the third requires that the causally active thing be extended [i.e., occupy a region of space]. Your notion of the soul entirely excludes extension, and it appears to me that an immaterial thing can't possibly touch anything else.
Also, if, as you say, thinking is the essential property of human souls, what about unborn children and people who have fainted, who presumably have souls without thinking?
René, May 21, 1643:
Admittedly in my writings I talk much more about the fact that the soul thinks than about the question of how it is united with the body. This idea of the union of the soul and the body is basic and can be understood only through itself. It's so easy to get confused by using your imagination or trying to apply notions that aren't appropriate to the case!
For a comparison, however, think about how the weight of a rock moves it downwards. One might (mistakenly, I hope later to show) think of weight as a "real quality" about which we know nothing except that it has the power to move the body toward the centre of the earth. The soul's power to move the body is analogous.
Elisabeth, June 10, 1643:
Please forgive my stupidity! I wish I had the time to develop your level of expertise. But why should I be persuaded that an immaterial soul can move a material body by this analogy to weight? If we think in terms of the old idea of weight, why shouldn't we then conclude by your reasoning that things move downward due to the power of immaterial causes? I can't conceive of "what is immaterial" except negatively as "what is not material" and as what can't enter into causal relations with matter. I'd rather concede that the soul is material than that an immaterial thing could move a body.
René, May 28, 1643:
This matter of the soul's union with the body is a very dark affair when it comes from the intellect (whether alone or aided by the imagination). People who just use their senses, in the ordinary course of life, have no doubt that the soul moves the body. We shouldn't spend too much time in intellectual thinking. In fact,
I never spend more than a few hours a day in the thoughts involving the imagination, or more than a few hours a year on thoughts that involve the intellect alone. I give all the rest of my time to the relaxation of the senses and the repose of the mind.
The human mind can't clearly conceive the soul's distinctness from the body and its union with the body simultaneously. The comparison with weight was imperfect, but without philosophizing everyone knows that they have body and thought and that thought can move the body.
But since you remark that it is easier to attribute matter and extension to the soul than to credit it with the capacity to move and be moved by the body without having matter, please feel free to attribute this matter and extension to the soul -- because that's what it is to conceive it as united to the body.
Still, once you do this, you'll find that matter is not thought because the matter has a definite location, excluding other matter. But again, thinking too much about metaphysics is harmful.
Elisabeth, July 1, 1643:
I hope my letters aren't troubling you.
I find from your letter that the senses show me that the soul moves the body, but as for how it does so, the senses tell me nothing about that, any more than the intellect and imagination do. This leads me to think that the soul has properties that we don't know -- which might overturn your doctrine... that the soul is not extended.
As you have emphasized in your writings, all our errors come from our forming judgments about things we don't perceive well enough. Since we can't perceive how the soul moves the body, I am left with my initial doubt, that is, my thinking that perhaps after all the soul is extended.
There is no record of a reply by Descartes.
---------------------------------------
Zing! Elisabeth shows up so much better than Descartes in this exchange. She immediately homes in on the historically most important (and continuing) objection to Cartesian substance dualism: the question of how, if at all, an immaterial soul and a material object could causally interact. She efficiently and elegantly formulates a version of the principle of "the causal closure of the physical", according to which material events can only be caused by other material events, connecting that idea both with Descartes' denial that the soul is extended in space and with the view, widely accepted by early modern philosophers before Newton, that physical causation requires direct physical contact (no "action at a distance"). Jaegwon Kim notes (2011, p. 49) that hers might be the first causal argument for a materialist view of the mind. To top it off, she poses an excellent objection (from fetuses and fainting spells) to the idea that thinking is essential to having a soul.
Descartes' reply by analogy to weight is weak. As Elisabeth notes, it doesn't really answer the question of how the process is supposed to work for souls. Descartes' own theory of weight (articulated the subsequent year in Principles of Philosophy, dedicated to Elisabeth) involves action by contact (light particles spinning off the rotating Earth shoot up, displacing heavier particles down: IV.20-24). At best, Descartes is saying that the false, old idea of weight didn't involve contact, so why not think souls can also have influence without contact? Elisabeth's reply implicitly suggests a dilemma: If downward motion is by contact, then weight is not an example of how causation without contact is possible. If downward motion is not by contact, then shouldn't we think (absurdly?) that things move down due to the action of immaterial souls? She also notes that "immaterial" just seems to be a negative idea, not something we can form a clear, positive conception of.
Elisabeth's response forces Descartes concede that we can't in fact think clearly and distinctly about these matters. This is a major concession, given the centrality of the standard of "clear and distinct" ideas to Descartes' philosophy. He comes off almost as a mysterian! He also seems to partly retract what is perhaps the most central idea in his dualist metaphysics -- that the soul does not have extension. Elisabeth should feel free to attribute matter and extension to the soul, after all! Indeed, in saying that attributing matter and extension is "what it is to conceive [the soul] as united to the body", Descartes seriously muddies the interpretation of his positive view about the nature of souls.
It's also worth noting that Descartes entirely ignores Elisabeth's excellent fetus and fainting question.
I had previously been familiar with Descartes' famous quote that he spends no more than a few hours a year on thoughts involving the intellect alone; but reading the full exchange provides interesting context. His aim in saying that is to convince Elisabeth not to put too much energy into objecting to his account of how the soul works.
Understandably, Elisabeth is dissatisfied. She even gestures (though not in so many words) toward Descartes' methodological self-contradiction: Descartes famously says that philosophizing requires that we have clear ideas and that our errors all arise from failure to do so -- yet here he is, saying that there's an issue at the core of his metaphysics about which it's not possible to think clearly! Shouldn't he admit, then, that on this very point he's liable to be mistaken?
If Descartes attempted a further reply, the reply is lost. Their later correspondence treats other issues.
The whole correspondence is just 15 pages, so I'd encourage you to read it yourself. This summary necessarily omits interesting detail and nuance. In this exchange, Elisabeth is by far the better philosopher.